Olivier Métais

Spectral large-eddy simulation of isotropic and stably stratified turbulence

We first recall the concepts of spectral eddy viscosity and diffusivity, derived from the two-point closures of turbulence, in the framework of large-eddy simulations in Fourier space. The case of a spectrum which does not decrease as

A numerical investigation of the coherent vortices in turbulence behind a backward-facing step

k^{-\frac{5}{3}}

Coherent structures in rotating three-dimensional turbulence

at the cutoff is studied. Then, a spectral large-eddy simulation of decaying isotropic turbulence convecting a passive temperature is performed, at a resolution of 128 3 collocation points. It is shown that the temperature spectrum tends to follow in the energetic scales a k −1 range, followed by a

Numerical experiments in forced stably stratified turbulence

k^{-\frac{5}{3}}

Direct and large-eddy simulation IV

inertial–convective range at higher wavenumbers. This is in agreement with previous independent calculations (Lesieur & Rogallo 1989). When self-similar spectra have developed, the temperature variance and kinetic energy decay respectively like t −1.37 and t −1.85 , with identical initial spectra peaking at k i = 20 and ∝ k 8 for k → 0. In the k −1 range, the temperature spectrum is found to collapse according to the law E T ( k, t ) = 0.1η(〈 u 2 〉/e) k −1 , where e and η are the kinetic energy and temperature variance dissipation rates. The spectral eddy viscosity and diffusivity are recalculated explicitly from the large-eddy simulation: the anomalous ∝ ln k behaviour of the eddy diffusivity in the eddy-viscosity plateau is shown to be associated with the large-scale intermittency of the passive temperature: the p.d.f. of the velocity component u is Gaussian (∼ exp − X 2 ), while the scalar T , the velocity derivatives ∂ u /∂ x and ∂ u /∂ z , and the temperature derivative ∂ T /∂ z are all close to exponential exp - | X | at high | X |. The pressure distribution is exponential at low pressure and Gaussian at high. For stably stratified Boussinesq turbulence, the coupling between the temperature and the velocity fields leads to the disappearance of the ‘anomalous’ temperature behaviour ( k −1 range, logarithmic eddy diffusivity and exponential probability density function for T ). These are the highest-resolution calculations ever performed for this problem. We also split the eddy viscous coefficients into a vortex and a wave component. In both cases (unstratified and stratified), comparisons with direct numerical simulations are performed. Finally we propose a generalization of the spectral eddy viscosity to highly intermittent situations in physical space: in this structure-function model , the spectral eddy viscosity is based upon a kinetic energy spectrum local in space. The latter is calculated with the aid of a local second-order velocity structure function. This structure function model is compared with other models, including Smagorinskys, for isotropic decaying turbulence, and with high-resolution direct simulations. It is shown that low-pressure regions mark coherent structures of high vorticity. The pressure spectra are shown to follow Batchelors quasi-normal law:

Vortex control of bifurcating jets: A numerical study

\alpha C^2_{\rm k}\epsilon^{\frac{4}{3}}k^{-\frac{7}{3}}

On the influence of coherent structures upon interscale interactions in turbulent plane jets

( C k is Kolmogorovs constant), with α ≈ 1.32.

Effects of spanwise rotation on the vorticity stretching in transitional and turbulent channel flow

This paper presents a statistical and topological study of a complex turbulent flow over a backward-facing step by means of direct and large-eddy simulations. Direct simulations are first performed for an isothermal two-dimensional case. In this case, shedding of coherent vortices in the mixing layer is demonstrated. Both direct and large-eddy simulations are then carried out in three dimensions. The subgrid-scale model used is the structure-function model proposed by Metais & Lesieur (1992). Lowstep computations corresponding to the geometry of Eaton & Johnstons (1980) laboratory experiment give turbulence statistics in better agreement with the experimental data than both Smagorinskys method and K -e modelling. Furthermore, calculations for a high step show that the eddy structure of the flow presents striking analogies with forced plane mixing layers: large billows are shed behind the step with intense longitudinal vortices strained between them.

Statistical Predictability of Decaying Turbulence

Numerical simulations investigating the formation and stability of quasi-twodimensional coherent vortices in rotating homogeneous three-dimensional flow are described. In a numerical study of shear flows Lesieur, Yanase & Metais (1991) found that cyclones (respectively anticyclones) with 1~02~1 - 0(2R), where (02D is the vorticity and Q is the rotation rate, are stabilized (respectively destabilized) by the rotation. A study of triply periodic pseudo-spectral simulations (64’) was undertaken in order to investigate the vorticity asymmetry in homogeneous turbulence. Specifically, we examine (i) the possible three-dimensionalization of initially two-dimensional vortices and (ii) the emergence of quasi-two-dimensional structures in initially-isotropic threedimensional turbulence. Direct numerical simulations of the Navier-Stokes equations are compared with large-eddy simulations employing a subgridscale model based on the second-order velocity structure function evaluated at the grid separation and with simulations employing hyperviscosity. Isolated coherent two-dimensional vortices, obtained from a two-dimensional decay simulation, were superposed with a low-amplitude three-dimensional perturbation, and used to initialize the first set of simulations. With R = 0, a threedimensionalization of all vortices was observed. This occurred first in the small scales in conjunction with the formation of longitudinal hairpin vortices with vorticity perpendicular to that of the initial quasi-two-dimensional flow. In agreement with centrifugal stability arguments, when 2R = [cuZD],,,,, a rapid destabilization of anticyclones was observed to occur, whereas the initial two-dimensional cyclonic vortices persisted throughout the simulation. At larger Q, both cyclones and anticyclones remained two-dimensional, consistent with the Taylor-Proudman theorem. A second set of simulations starting from isotropic three-dimensional fields was initialized by allowing a random velocity field to evolve (52 = 0) until maximum energy dissipation. When the simulations were continued with 252 = [o . n],,,,, /Q, the three-dimensional flow was observed to organize into two-dimensional cyclonic vortices. At larger Q, two-dimensional anticyclones also emerged from the initially-isotropic flow. The consequences for a variety of industrial and geophysical applications are clear. For quasi-two-dimensional eddies whose characteristic circulation times are of the order of Q-’, rotation induces a complete disruption of anticyclonic vortices, while stabilizing cyclonic ones.

Rotating free-shear flows. Part 2. Numerical simulations

We present results of numerical simulations of stably stratified, randomly forced turbulence. The selection of forcing and damping are designed to give insight into the question of whether cascade of energy to large scales is possible for strongly stratified three-dimensional turbulence in a manner similar to two-dimensional turbulence. We consider narrow-band wavenumber forcing, whose angular distribution ranges from two-dimensional to three-dimensional isotropic. Our principal results are as follows; for two-dimensional forcing, and for sufficiently small Froude number, the statistically steady state is characterized by a weakly inverse-cascading horizontal-velocity variance field. The vertical variability of the horizontal-velocity field is pronounced, but seems to approach a limit independent of the Brunt–Vaisala frequency N , as N → ∞. If, on the other hand, the Froude number exceeds a critical value, the vertical variability is weak, and the statistics of the scales larger than the forcing scale is near that predicted by inviscid equipartitioning. For all forcing functions considered the vertical motion and temperature field ( w , T ), centred at smaller scales, are more three-dimensionally isotropic, with no large-scale organization. At large N , (small Froude number) the w -field scales as 1/ N , with horizontal motion field nearly independent of N . Furthermore, at large N and for horizontal forcing, the horizontal motion field is consistent with the condition that a substantial fraction of the total dissipation is attributable to an effective drag acting upon all horizontal scales of motion, which in turn flattens the slope of the energy spectrum in the inverse-cascade range, and increases it in the enstrophy-cascade range.